GG 711: Advanced Techniques in Geophysics and Materials Science

Pavel ZiniHIGP, University of Hawaii, Honolulu, USA

Lecture 15

Measurements of the Elastic Properties of Minerals and Functional Materials

www.soest.hawaii.edu\~zinin

Seismological images of theEarth's mantle reveal three distinctchanges in velocity structure, atdepths of 410, 660 and 2,700 km.The first two are best explained bymineral phase transformations,whereas the third—the D" layer—probably reflects a change inchemical composition and thermalstructure.

Lay et al., Physics Today, 1990

Cross Sections of the Earth and Its Elastic Properties

Average elastic parameters as afunction of depth. The P-wavevelocity Vp, S-wave velocity VS; anddensity p are determined fromseismological analysis. The figure isbased on the Primary Reference EarthModel (PREM). PREM was created in1981 (Dziewonski and Anderson,Phys. Earth. Planet. Int., 25, 297).

Elasticity in Geophysics

Understanding of the elastic behavior of minerals under highpressure is a crucial factor for developing a model of the Earth’sstructure because most information about the Earth’s interior comesfrom seismological data.

Seismological studies give us a high-definition 3-D picture of theEarth’s interior in terms of seismic velocity and density. Near thesurface, observations of these properties can be compared with rocksamples. As we go deeper into the Earth, interpretation of seismicdata is more difficult.

Laboratory measurements of velocities and other elastic propertiesof minerals are the key to understanding this seismic information,allowing us to translate it into quantities such as chemicalcomposition, mineralogy, temperature, and preferred orientation ofminerals.

Example: An architect wants todesign a 5m high circular pillarwith a radius of 0.5 m that holds abronze statue that weighs 1.004 kg.He chooses concrete for thematerial of the pillar (E=10 GPa).How much does the pillarcompress?

5m

Elasticity in Materials Science and Engineering

The Chrysler Building is an Art Decoskyscraper in New York City, locatedon the east side of Manhattan in theTurtle Bay area at the intersection of42nd Street and Lexington Avenue.Standing at 319 metres, it was theworld's tallest building for 11 monthsbefore it was surpassed by the EmpireState Building in 1931.

Elasticity in Materials Science and Engineering

Elasticity in Materials Science and Engineering

The Tacoma Narrows Bridge is a pair of mile-long suspension bridges in the U.S. state ofWashington, which carry State Route 16 across the Tacoma Narrows between Tacoma and theKitsap Peninsula. The 1940 collapse of Tacoma Narrows Bridge (1940), as the original TacomaNarrows Bridge is known, is sometimes characterized in physics textbooks as a classical exampleof resonance; although, this description is misleading. The catastrophic vibrations that destroyedthe bridge were not due to simple mechanical resonance, but to a more complicated oscillationbetween the bridge and winds passing through it, known as aeroelastic flutter.

Acoustical images aredifficult for directinterpretation: Ultrasoundimages of a fetus duringseventeen of development(left) and an artist’srendering of the image.(after Med. Encyclopedia,2005)

Elasticity in Medicine

“By striving to do the impossible, man has always achieved what is possible.” Bakunin

In mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extensionof a spring is in direct proportion with the load added to it as long as this load does not exceed theelastic limit. Materials for which Hooke's law is a useful approximation are known as linear-elastic or"Hookean" materials (Wikipedia, 2009). Mathematically, Hooke's law states that

Where x is the displacement of the end of the spring from its

equilibrium position;F is the restoring force exerted by the material; andk is the force constant (or spring constant).

When this holds, the behavior is said to be linear. There is anegative sign on the right hand side of the equation because therestoring force always acts in the opposite direction of thedisplacement (for example, when a spring is stretched to the left,it pulls back to the right).

kxF

Hooke’s Law

In 1660, Hooke discovered the law of elasticity which bears his name. He first described this discovery in theanagram "ceiiinosssttuv", whose solution he published in 1678 as "Ut tensio, sic vis" meaning "As the extension, sothe force." Hooke's work on elasticity culminated, for practical purposes, in his development of the balance spring orhairspring, which for the first time enabled a portable timepiece - a watch - to keep time with reasonable accuracy.

Robert Hooke

Much has been written about the unpleasant side of Hooke'spersonality, starting with comments by his first biographer, RichardWaller, that Hooke was "in person, but despicable" and "melancholy,mistrustful, and jealous." Waller's comments influenced otherwriters for well over two centuries, so that a picture of Hooke as adisgruntled, selfish, anti-social curmudgeon dominates many olderbooks and articles. (Wikipedia, 2009).

Portrait of Hooke

by history painter Rita Greer, 2004.

No authenticated portrait of Robert Hooke exists, a situation sometimesattributed to the heated conflicts between Hooke and Isaac Newton. InHooke's time, the Royal Society met at Gresham College, but within afew months of Hooke's death Newton became the Society's president andplans were laid for a new meeting place. When the move to new quartersfinally was made a few years later, in 1710, Hooke's Royal Societyportrait went missing, and has yet to be found.

Lacking money to support himself. Hooke worked as a part-time assistant of Robert Boyle. It worked our sowell that he never completed life degree, becoming instead the first full-time, paid, professional scientist.While they were working together, Hook discovered the law chat relates the pressure and volume of a gas.Boyle wrote about this in a book published in tool, and was careful always to cay that it was Hooke‘sdiscovery. But because the discovery appeared its Boyle's book, to this day is known as "Boyle's Law“. In thesame year, 1661, a group of men, including Robert Boyle, formed a society "for the promoting ofExperimental Philosophy" under the King‘s patronage. This became the Royal Society. Hooke became its first"Curator of Experiments." His job was to demonstrate and test new discoveries, either those made by theFellows of the Society or by himself, or reported from other places.

Atomic Forces and Elasticity in Solids

amorphous ordered crystalline

Modulus of Elasticity and the Interatomic Potential

• Recall that energy between atoms depends on their separation• Recall also thatMinimum in energy zero net force

tensioncompression

Applying tension or compression raises energy of material

)/( 3mkgVM

density

The bond force curve rEF

Stress and Strain Definitions

An elastic modulus, or modulus of elasticity, is the mathematical description of anobject or substance's tendency to be deformed elastically (i.e., non-permanently) whena force is applied to it. The elastic modulus of an object is defined as the slope of itsstress-strain curve in the elastic deformation region:

Where E is the elastic modulus; stress is the force causing the deformation divided bythe area to which the force is applied; and strain is the ratio of the change caused by thestress to the original state of the object.

If stress is measured in Pascals (Pa = N/m2), since strain is a unitless ratio, then the unitsof E are Pa as well.

E

Some Definitions

Specifying how stress and strain are to be measured, including directions, allows for many types ofelastic moduli to be defined. The three primary ones are:

* Young's modulus (E) describes tensile elasticity, or the tendency of an object to deform along anaxis when opposing forces are applied along that axis; it is defined as the ratio of tensile stress totensile strain. It is often referred to simply as the elastic modulus.

* The shear modulus or modulus of rigidity (G or µ) describes an object's tendency to shear (thedeformation of shape at constant volume) when acted upon by opposing forces; it is defined as shearstress over shear strain. The shear modulus is part of the derivation of viscosity.

* The bulk modulus (K) describes volumetric elasticity, or the tendency of an object to deform inall directions when uniformly loaded in all directions; it is defined as volumetric stress overvolumetric strain, and is the inverse of compressibility. The bulk modulus is an extension of Young'smodulus to three dimensions.

Three other elastic moduli are Poisson's ratio, Lamé's first parameter, and P-wave modulus.Homogeneous and isotropic (similar in all directions) materials (solids) have their (linear) elasticproperties fully described by two elastic moduli, and one may choose any pair. Given a pair of elasticmoduli, all other elastic moduli can be calculated according to formulas in the table below at the endof page.Inviscid fluids are special in that they cannot support shear stress, meaning that the shearmodulus is always zero. This also implies that Young's modulus is always zero.

Modulus of Elasticity: Young’s modulus

E – is Young’s elastic modulus: theratio of stress to strain (the measureof resistance to elastic deformation).

Young's modulus is named after Thomas Young, the 19thcentury British scientist. However, the concept wasdeveloped in 1727 by Leonhard Euler, and the firstexperiments that used the concept of Young's modulus in itscurrent form were performed by the Italian scientistGiordano Riccati in 1782 — predating Young's work by 25years (Wikipedia, 2009).

F F

o

FA

Cross-sectional area Ao

LL0 E

0L L

Young’s moduli of Some Materials

E – is Young’s elastic modulus: the ratio of stress to strain.

E

Material Young’s modulus (GPa)

Rubber (small strain) 0.01-0.1

wood 1-10

bone 9-16

concrete 20

steel 200

Poisson’s Ratio

z

y

z

x

Poisson's ratio, named after Siméon Poisson, is the ratio, when a sample object is stretched, of thecontraction or transverse strain (perpendicular to the applied load), to the extension or axial strain(in the direction of the applied load).

When a sample cube of a material is stretched in one direction, it tends to contract (or occasionally,expand) in the other two directions perpendicular to the direction of stretch. Conversely, when asample of material is compressed in one direction, it tends to expand (or rarely, contract) in theother two directions. This phenomenon is called the Poisson effect. Poisson's ratio () is a measureof the Poisson effect.

Elastic dimensional change will occur transverse to applied uniaxial load:

Shear Modulus

//

xy

xyz

F A FhGx h A x

wherexy = F/A , = shear stress;F is the force which actsA is the area on which the force actsxy = x/I = tan = shear strain;x is the transverse displacementh is the initial height

In materials science, shearmodulus or modulus of rigidity,denoted by G, or sometimes µ, isdefined as the ratio of shear stressto the shear strain:

Shear modulus is usually measured in GPa (gigapascals)

h

Bulk Modulus

VPVK

where P is pressure, V is volume, and dP/dVdenotes the partial derivative of pressure withrespect to volume. The inverse of the bulkmodulus gives a substance's compressibility.

The bulk modulus (K) of a substancemeasures the substance's resistance touniform compression. It is defined as thepressure increase needed to cause a givenrelative decrease in volume. Its base unit isPascal. The bulk modulus K can beformally defined by the equation:

All moduli describe the material's response (strain) to specific kinds of stress: the shearmodulus describes the response to shear, and Young's modulus describes the response tolinear strain. For a fluid, only the bulk modulus is meaningful. For an anisotropic solidsuch as wood or paper, these three moduli do not contain enough information todescribe its behaviour, and one must use the full generalized Hooke's law.

Compressibility: 1/K (Bulk modulus)

The deformation of solids

For small stress, strain and stress are linearly correlated.

Strain = Constant*StressConstant: elastic modulusThe elastic modulus depends on:• Material that is deformed•Type of deformation (a different modulus is defined for different types of deformations)

Materials Bulk Modulus (GPa)

Water 2.2

Glass 35 to 55

Steel 160

Diamond 442

Beyond Hooke’s law

Beyond the elastic limit an object is permanently deformed (it does not return to its original shape if the stress is removed).

Understand the difference between elastic and plastic deformation

• Know how to determine mechanical properties from the results of a tensile test– Elastic modulus • Yield strength– Tensile strength • Strain to failure

• Understand how the mechanical properties of ceramics differ from those of ductile metals

Tensile strength

Tensile strength is indicated by themaxima of a stress-strain curve and, ingeneral, indicates when necking willoccur. As it is an intensive property, itsvalue does not depend on the size of thetest specimen. It is, however, dependenton the preparation of the specimen andthe temperature of the test environmentand material.

1. Ultimate Strength2. Yield Strength3. Rupture4. Strain hardening region5. Necking region.A: Apparent (Engineering) stress (F/A0)B: Actual (True) stress (F/A) Stress vs. Strain curve typical of structural steel

There are three definitions of tensile strength:

Stress vs. Strain curve typical of structural steel

The yield strength or yield point of amaterial is defined in engineering andmaterials science as the stress at which amaterial begins to deform plastically. Priorto the yield point the material will deformelastically and will return to its originalshape when the applied stress is removed.Once the yield point is passed somefraction of the deformation will bepermanent and non-reversible (Wikipedia,2009).

Yield strength

True elastic limit, Proportionality limit, Elastic limit (yield strength)

Stress vs. Strain curve typical of structural steel

Ultimate strength: maximum force per unitarea a material can withstand before itbreaks or fractures.

Different for compression and tension.

Ultimate strength

Ultimate Strength of MaterialsMaterials Tensile

Strength (N/m2)

Compressive Strength (N/m2)

Iron 1.7 x 108 5.5 x 108

Steel 5.0 x 108 5.0 x 108

Bone 1.2 x 108 1.5 x 108

Marble - 8.0 x 107

Brick 1 x 106 3.5 x 107

Concrete 2 x 106 2.0 x 107

Tensile strength

Stress vs. Strain curve typical of aluminum1. Ultimate Strength2. Yield strength3. Proportional Limit Stress4. Rupture5. Offset Strain (typically 0.2%).

After a metal has been loaded to its yield strength itbegins to "neck" as the cross-sectional area of thespecimen decreases due to plastic flow. Whennecking becomes substantial, it may cause areversal of the engineering stress-strain curve,where decreasing stress correlates to increasingstrain because of geometric effects.

The peak stress on the engineering stress-straincurve is known as the ultimate strength. After aperiod of necking, the material will rupture and thestored elastic energy is released as noise and heat.The stress on the material at the time of rupture isknown as the breaking strength.

Ductile metals do not have a well defined yieldpoint. The yield strength is typically defined by the"0.2% offset strain". The yield strength at 0.2%offset is determined by finding the intersection ofthe stress-strain curve with a line parallel to theinitial slope of the curve and which intercepts theabscissa at 0.2%. A stress-strain curve typical ofaluminium along with the 0.2% offset line is shownin the figure below.

Callister Fig. 6.11

Plastic behavior in a tensile testTensile strength (M) (a.k.a. ultimate tensile strength)

Mechanical Properties of Metals

Yielding

Fracture (F)

Necking

Lecture 26, summer 2007Mechanical Properties I: Metals & Ceramics

ENGR 145, Chemistry of MaterialsCase Western Reserve University

Callister Fig. 6.13

Two measures of ductility:% elongation

% area reduction

0

0

% 100fl lEL

l

0

0

% 100fA ARA

A

Brittle vs. ductile

Mechanical Properties of Metals — Plastic Behavior

Typical Mechanical Properties of Several Metals and Alloys in a Annealed State

Metal Alloy Yield Strength (MPa)

Tensile Strength (MPa)

Ductility %EL

Aluminum 35 90 40Copper 69 200 45Iron 130 262 45Nickel 138 480 40Titanium 450 520 25Molybdenum 565 655 35

Hardness

Hardness refers to various properties of matter in the solid phase that give it high resistance to various kinds of shape change when force is applied. Hard matter is contrasted with soft matter.

Macroscopic hardness is generally characterized by strong intermolecular bonds. However, the behavior of solid materials under force is complex, resulting in several different scientific definitions of what might be called "hardness" in everyday usage.

In materials science, there are three principal operational definitions of hardness:

* Scratch hardness: Resistance to fracture or plastic (permanent) deformation due to friction from a sharp object

* Indentation hardness: Resistance to plastic (permanent) deformation due to a constant load from a sharp object

* Rebound hardness: Height of the bounce of an object dropped on the material, related to elasticity.

What is Indentation test

• Easy and cheap to perform: • Properties:

– Elastic properties : Modulus– Plastic properties: Hardness– Time dependent properties: Creep

Hardness

In a traditional indentation test (macro or micro indentation), a hard tip whose mechanicalproperties are known (frequently made of a very hard material like diamond) is pressed into asample whose properties are unknown. The load placed on the indenter tip is increased as thetip penetrates further into the specimen and soon reaches a user-defined value. At this point, theload may be held constant for a period or removed. The area of the residual indentation in thesample is measured and the hardness, H, is defined as the maximum load, Pmax, divided by theresidual indentation area, Ar, or

rAPH max

An AFM image of an indent left by aBerkovich tip in a Zr-Cu-Al metallic glass;the plastic flow of the material around theindenter is apparent (Wikipedia, 2009).

Brittle vs. ductile

0 100 200 300 4000

20

40

60

80

100 (c)

(b)

(a)

purely elasticunloading-reloading pristine material

Indenter Displacement (nm)

App

lied

Load

(mN

)

nanoindentation

B4C100 mN1 mN/sec

220 240 260 280 300 320

10

20

30

40

Con

tact

Pre

ssur

e (G

Pa)

Contact Depth (nm)

400 600 800 1000 1200 1400 1600 1800 2000

In

tens

ity (a

.u.)

Wavenumber (cm-1)

1 m1 m

A typical load-displacement curve in the Berkovich nanoindentation of boron carbide. (b) The correspondingunloading and reloading average contact pressure vs. contact depth curves. The poressure at maximum load reaches40 GPa. (c) Raman spectra of a 100 mN Berkovich nanoindent (SEM image, inset) and the pristine material showingstructural changes induced by extremely high contact pressures associated with indentation. From Ref. (Domnich etal., 2001)

Vickers hardness vs. shear modulus for various crystalline materials: c-BC2N, nc-TiN/a-Si3N4 , nc-TiN/a-BN

100 200 300 400 500 600

20

40

60

80

100

n c -T iN /a -S i3 N 4 n c -T iN /a -B N

TiNAl2O3

SiC

B4CSi3N4

B6OTB2

c-BC2N

cBN

Vic

kers

Har

dnes

s (G

Pa)

Shear Modulus (GPa)

Diamond

From M. H. Manghnani, P. V. Zinin, S. N. Tkachev, P. Karvankova, S. Veprek. “Mechanical Properties and Hardness of Advanced Superhard Nanocrystalline films and nanomaterials”. In Proceedings of the 6th Pacific Rim Conference on Ceramic and Glass Technology.Y Gogotsi ed. Wiley (2006), p. 155-160.

Measurements of the Elastic Properties of Materials by Ultrasonic techniques

P and S waves

Transverse Waves - The particles of the medium undergo displacements in a direction perpendicular to the wave velocity.

Longitudinal (Compression) Waves - The particles of the medium undergo displacements in a direction parallel to the direction of wave motion.

Surface acoustic waves

A surface acoustic wave (SAW) is an acoustic wave traveling along thesurface of a material exhibiting elasticity, with an amplitude thattypically decays exponentially with depth into the substrate.

Confocal Acoustic MicroscopeThe acoustic microscope was developed as a tool for studying the internal microstructure ofnontransparent solids or biological materials. In acoustic microscopy, a sample is imagedby ultrasound waves, and the contrast in reflection furnishes a map of the spatialdistribution of the mechanical properties

The schematic diagram of the combined optical and acoustic microscope (Weiss, Lemor et al., IEEE Trans. Ultrason. Ferroelectr. Freq. Contr., 54 2257, 2007).Right: A photograph of the combined optical (Olympus IX81) and time-resolved scanning acoustic microscope, SASAM, Fraunhofer-Institute for Biomedical Technology, St. Ingbert, Germany.

Sound Velocity by Time –Resolved Microscope

1 0 1t t

The setup for the quantitative time-resolved acoustic microscopy

1

1 0 1

2Wcd

t t

12

12 2 1

2dc

t t

Step 1

Step 2

Echo signals from the HeLa cell

20 25 30 35 40 45 50

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

A2

t2

Am

plitu

de (V

)

Time delay (ns)

t1A1

(a)

20 25 30 35 40 45 50-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

Ao

to (b)

Am

plitu

de (V

)

Time delay (ns)

Echo signal from the glass substrate.

1 1

1 22 1 1

41 log2

o s o s oo e

s s oo

A Z Z Z Z Z Zd A Z Z Z ZZ Z

Sound Attenuation in Cells

11

oo

o

A AZ ZA A

1

1Zc

Position phase Thickness (µm)

Velocity (m/s)

Densityg/cm3

Attenuation (Neper/μm)

A Beforedivision 12.58 1550 1.239 0.0201

B Afterdivision 15.57 1551 1.120 0.0308

C Afterdivision 11.93 1548 1.306 0.0297

D Normal 5.27 1501 1.184 0.0209

Sound Attenuation during HeLa Cells Division

Generation of Acoustical Waves by Laser

When an ultra-short laser pulse,known as the pump pulse, isfocused onto an opaque surface.The optical absorption results in athermal expansion that launchesan elastic strain pulse. This strainpulse mainly consists oflongitudinal acoustic waves thatpropagate directly into the bulk.

A schematic of the geometry of a non-transparent sample excited by a laser source

The stress p’ producing in the medium is given by laser heating

Tcp Too 2'where T is the temperature rise, o is the density of the medium, co is the sound velocity

in the medium, αT is the linear thermal expansion coefficient, and T is the temperature change (Karabutov).

Elasticity in Geophysics

Dispersion curves measuredby the laser-SAW methodfor different DLC coatingthicknesses.

Sketch of the Laser-SAW technique

Thickness (m)

K (GPa)

(GPa)

E(GPa)

E (GPa)Laser-SAW

Cr-DLC/steel

2.90 28.8 31.3 68.9 65.4

DLC/silicon

0.495 487 200.1 462 457

Derivation of Properties from Velocities: Labels of seismic waves which penetrated to the mantle, core or inner core. The stippling indicates possible complexity in the upper mantle.

Elasticity in Geophysics

Divisions in the Earth's MantleWith increasing pressure, uppermantle minerals such as(Mg,Fe)2SiO4 olivine, garnet,and pyroxene transform todenser mineral phases. Thesetransformations are associatedwith major jumps in seismicvelocity, called discontinuities,at depths of 410 and 660 km.

Transformations to a perovskite-structured phase with theformula (Mg,Fe)SiO3 isgenerally associated with themajor seismic discontinuity at660 km depth. This phase, alongwith (Mg,Fe)O and perovskite-structured CaSiO3, are thought toaccount for the bulk of the lowermantle deeper than 660 km.

Mineral volume fractions for the top 1000 km of a pyrolite mantle. Pyrolite is a theoretical rock consideredto be the best approximation of the composition of Earth's upper mantle (from Frost, Elements, 2008 ).

Bernal was the first to propose that rapid increases in seismic velocity in the mantlemight be due to phase transformations rather than a change in composition.Experiments in the mid-1960s showed that the olivine component of peridotiteundergoes successive pressure-dependent transformations to the spinel structure(ringwoodite), and ultimately breaks down to form (Mg,Fe)SiO3 perovskite plus(Mg,Fe)O :

(Mg,Fe)2SiO4 = (Mg,Fe)2SiO4 Pressure, 13-14 GPa; depth, 410 km

Olivine Wadsleyite

(Mg,Fe)2SiO4 = (Mg,Fe)2SiO4 Pressure, 18 GPa; depth, 520 km

Wadsleyite Ringwoodite

(Mg,Fe)2SiO4 = (Mg,Fe)SiO3 + (Mg,Fe)O Pressure, 23 GPa; depth, 660 km

Ringwoodite Perovskite Magnesiowustite

Phase Transformations in the Mantle

Elastic Properties of Minerals

The variation in sound velocity with depth for various key mantle minerals: olivine (Ol), diopside (Cpx),enstatite (Opx), garnet (Gt), majorite-garnet solid solution (Mj50), wadsleyite (Wd), ringwoodite (Rw),magnesiowüstite (Mw), Mg-silicate perovskite (Pv). Increases in temperature for an adiabatic gradientare taken into account. The reference model PREM (Dziewonski and Anderson 1981) is shown forreference. The length of an adiabat indicates approximately the maximum pressure stability of any givenphase. (From Bass, Elements, 2008 ).

Perovskite

A perovskite is any material with the same type of crystal structure as calcium titaniumoxide (CaTiO3), known as the perovskite structure. Perovskites take their name from thiscompound, which was first discovered in the Ural mountains of Russia by Gustav Rose in1839 and is named after Russian mineralogist, L. A. Perovski (1792-1856).

A phase transition of MgSiO3 perovskite, the most abundant component of the lowermantle, to a higher-pressure (125 GPa, 2000K) form called post-perovskite wasrecently discovered for pressure and temperature conditions in the vicinity of theEarth’s core–mantle boundary.

Crystal structure of the post-perovskite phase of(Mg,Fe)SiO3. The structure consists of layers oflinked silicon octahedra (yellow). Red spheres atvertices of SiO6 octahedra are oxygen ions, andblue spheres are magnesium and iron ions.

Discovery of Post Perovskites: Murakami et al., Nature 2004

Shear wave velocities of MgSiO3 perovskite andpostperovskite phase as a function of pressure at 300 K.Open circles show the data of post-perovskite phase in thepresent study, and filled circles those of perovskite(Murakami et al., Earth Planet. Sci. Lett. 2007).

Shear waves in the diamond-anvil cell

(A) The P-to-S acoustic wave converter (between a pair of tweezers) before sputteringthe P-transducer. The gem is an oriented single crystal of yttrium aluminum garnet,hand-faceted to tolerances of ±0.1°. It produces pure-mode elastic shear energy with1- to 10-μm wavelengths and well defined polarization direction for high-pressureelasticity experiments in the DAC. (B) Shear waves are introduced into the DACthrough one of the anvils (Jacobsen et al., PNAS 2004).

Laser ultrasonics (LU) in diamond anvil cells (LU-DAC)

Probe and pump lasers are on the same sides.

2

L

hc

is the time of flight (of sound pulse),c is the sound velocity and h is the sample thickness

Laser Ultrasonics (LU) in Diamond Anvil Cells (LU-DAC)

2 21 4d hc

The time delay for the arrival of theLL and TT echoes ss is equal to

α = L,T

2 24s c h

if we introduce following variables,

s = d2, = 2,

then the equation above can be rewritten

LU-DAC, point-source - point-receiver technique: sound velocities can be determined from the linear fitting of the experimental data in (s, ) coordinates.

Measurements of Longitudinal and Shear Wave Velocitiesin Iron by LU-DAC

The signals measured at different distances d. The step of the scan is 7.4 µm. The topsignal was measured at d = 43.6 µm. Pressure was 10.9 GPa.

0 10 20 30 40

-8

-7

-6

-5

-4

-3

-2

-1

0

1

SLFe-TT

TTLTLL

SLFe

STD

SLD

(ns)

Peaks P1 can be attributedto the arrival, with the timedelay LL after thepropagation in iron, of theLL wave that is excited asthe longitudinal (L) waveat the diamond/ironinterface and is reflectedby the iron/diamondinterface as thelongitudinal (L) wave.Peak TT is attributed toarrival of the transverse-transverse (TT) wave withtime delay TT, and the P2peak with time delayLT=TL is due to LT and TLacoustic mode conversionat the rear surface of theiron layer.

Longitudinal and Shear Wave Velocities in Iron by LU-DAC

Fitting of the SLD, STD, SLFe LL, LT/TL and SLFe-TT wave arrivals at 10.9 GPa.Thickness of the sample is taken from LL measurement to fit LT and TT peaks.

0 5 10 15 20 25 30 35 40

40

50

60

70

80

90

100

110

120

130 SLFe-TTSLD STD

SLFe

LL LT

Dis

tanc

e (

m)

(ns)

TT

Fitting of the P1, P2 and P3peaks at 22 GPa.

where δ=dL / d, =(h/d), =LT cL /d, q=cT /cL

2 22 2LL

LTL T

d d hd hc c

2 2 2 2

L L

L L T L

d d dc d h c d d h

2 2

4 3 2 22 2

-2 + 1+ -2 + =01 1q q

0 5 10 15 20 25 30

3

4

5

6

7

8

9

Vel

ocity

(km

/s)

Pressure (GPa)

Measurements of Longitudinal and Shear Wave Velocitiesin Iron by LU-DAC

Experimentally measured velocities of longitudinal (filled squares) and shear (filledtriangles) waves in pure iron layer as a function of pressure and those taken from literature.

Future Work

The presented work is a starting point in a new direction: study of elasticity ofthe non-transparent minerals such as iron, iron alloys (FexSi1-x, FexC1-x) andiron sulfides (FeS, FeS2) as well as iron-rich post-perovskites (Fe,Mg)SiO3under pressures above 30 GPa.Experimental data thus obtained using LU-DAC will shed light on the natureand composition of the D layer and earth’s core, and possibly on the formationof the Hawaiian Islands through the hot spot (Mao et al., PNAS, 2004).

Home Reading

1. Callister, Materials Science and Engineering: An Introduction (2003)

2. P. Zinin and W. Weise, “Theory and applications of acoustic microscopy”, in T. Kundu ed., Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization. CRC Press, Boca Raton, chapter 11, 654-724 (2004).

General Reference: M. Levy, H. Bass, R. Stern, V. Keppens eds., Handbook of ElasticProperties of Solids, Liquids, and Gases. Vol. I: Dynamical Methods for Measuringthe Elastic Properties of Solids, Academic Press, New York, 187-226 (2001).